Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 15, 2020

Special functions and Gauss–Thakur sums in higher rank and dimension

  • Quentin Gazda ORCID logo EMAIL logo and Andreas Maurischat ORCID logo


Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss–Thakur sums, as shown by Anglès and Pellarin. For Drinfeld–Hayes modules, generalizations of Anderson generating functions have been introduced by Green–Papanikolas and – under the name of “special functions” – by Anglès, Ngo Dac and Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. We also undertake the study of Gauss–Thakur sums for Anderson A-modules, and show that the result of Anglès–Pellarin relating values of the special functions to Gauss–Thakur sums holds in this generality.


Both authors thank Rudy Perkins for a result on the tensor powers of the Carlitz module, that finally does not appear anymore in the paper, but led us on the right track. This work is part of the PhD thesis of the first author under the supervision of Federico Pellarin.


[1] G. W. Anderson, t-motives, Duke Math. J. 53 (1986), no. 2, 457–502. 10.1215/S0012-7094-86-05328-7Search in Google Scholar

[2] G. W. Anderson, Rank one elliptic A-modules and A-harmonic series, Duke Math. J. 73 (1994), no. 3, 491–542. 10.1215/S0012-7094-94-07321-3Search in Google Scholar

[3] G. W. Anderson and D. S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), no. 1, 159–191. 10.2307/1971503Search in Google Scholar

[4] B. Anglès, T. Ngo Dac and F. Tavares Ribeiro, Special functions and twisted L-series, J. Théor. Nombres Bordeaux 29 (2017), no. 3, 931–961. 10.5802/jtnb.1007Search in Google Scholar

[5] B. Anglès and F. Pellarin, Universal Gauss–Thakur sums and L-series, Invent. Math. 200 (2015), no. 2, 653–669. 10.1007/s00222-014-0546-8Search in Google Scholar

[6] G. Böckle and U. Hartl, Uniformizable families of t-motives, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3933–3972. 10.1090/S0002-9947-07-04136-0Search in Google Scholar

[7] S. Bosch, Lectures on formal and rigid geometry, Lecture Notes in Math. 2105, Springer, Cham 2014. 10.1007/978-3-319-04417-0Search in Google Scholar

[8] A. El-Guindy and M. A. Papanikolas, Identities for Anderson generating functions for Drinfeld modules, Monatsh. Math. 173 (2014), no. 4, 471–493. 10.1007/s00605-013-0543-9Search in Google Scholar

[9] D. Goss, Basic structures of function field arithmetic, Ergeb. Math. Grenzgeb. (3) 35, Springer, Berlin 1996. 10.1007/978-3-642-61480-4Search in Google Scholar

[10] N. Green, Tensor powers of rank 1 drinfeld modules and periods, J. Number Theory (2019), 10.1016/j.jnt.2019.03.016. 10.1016/j.jnt.2019.03.016Search in Google Scholar

[11] N. Green and M. A. Papanikolas, Special L-values and shtuka functions for Drinfeld modules on elliptic curves, Res. Math. Sci. 5 (2018), no. 1, Paper No. 4. 10.1007/s40687-018-0122-8Search in Google Scholar

[12] U. Hartl and A.-K. Juschka, Pink’s theory of hodge structures and the hodge conjectures over function fields, Proceedings on the conference on “t-motives: Hodge structures, transcendence and other motivic aspects, European Mathematical Society, Zürich (2020), 31–182. 10.4171/198-1/2Search in Google Scholar

[13] R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar

[14] H. Matsumura, Commutative ring theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge University, Cambridge 1989. Search in Google Scholar

[15] A. Maurischat, Periods of t-modules as special values, J. Number Theory (2018), 10.1016/j.jnt.2018.09.024. 10.1016/j.jnt.2018.09.024Search in Google Scholar

[16] J. Neukirch, Algebraic number theory, Grundlehren Math. Wiss. 322, Springer, Berlin 1999. 10.1007/978-3-662-03983-0Search in Google Scholar

[17] F. Pellarin, Aspects de l’indépendance algébrique en caractéristique non nulle, Séminaire Bourbaki. Vol. 2006/2007, Astérisque 317, Société Mathématique de France, Paris (2008), 205–242, Exp. No. 973. Search in Google Scholar

[18] F. Pellarin, Values of certain L-series in positive characteristic, Ann. of Math. (2) 176 (2012), no. 3, 2055–2093. 10.4007/annals.2012.176.3.13Search in Google Scholar

[19] J.-P. Serre, Endomorphismes complètement continus des espaces de Banach p-adiques, Publ. Math. Inst. Hautes Études Sci. 12 (1962), 69–85. 10.1007/978-3-642-37726-6_55Search in Google Scholar

[20] S. K. Sinha, Periods of t-motives and transcendence, Duke Math. J. 88 (1997), no. 3, 465–535. 10.1215/S0012-7094-97-08820-7Search in Google Scholar

[21] D. S. Thakur, Gauss sums for 𝐅 q [ T ] , Invent. Math. 94 (1988), no. 1, 105–112. 10.1007/BF01394346Search in Google Scholar

[22] D. S. Thakur, Gamma functions for function fields and Drinfel’d modules, Ann. of Math. (2) 134 (1991), no. 1, 25–64. 10.2307/2944332Search in Google Scholar

[23] D. S. Thakur, Gauss sums for function fields, J. Number Theory 37 (1991), no. 2, 242–252. 10.1016/S0022-314X(05)80040-XSearch in Google Scholar

[24] D. S. Thakur, Behaviour of function field Gauss sums at infinity, Bull. Lond. Math. Soc. 25 (1993), no. 5, 417–426. 10.1112/blms/25.5.417Search in Google Scholar

[25] D. S. Thakur, Shtukas and Jacobi sums, Invent. Math. 111 (1993), no. 3, 557–570. 10.1007/BF01231298Search in Google Scholar

Received: 2020-08-24
Published Online: 2020-10-15
Published in Print: 2021-04-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 25.2.2024 from
Scroll to top button